The fact that a moduli space of curves like \({\mathcal {M}}_{g}\)  for \(g=1,2,3\) 
maps generically finitely to a Shimura variety helps a
lot for understanding the cohomology of local systems, and hence understanding
the behavior of quantities
like \(\# {\mathcal {M}}_{g,n}({\mathbb {F}}_q)\) . Things become much harder
if one does not know a priori which
modular forms or motives may show up.
The case of \({\mathcal {M}}_{4}\)  and \({\mathcal {M}}_{4,n}\)  illustrates this. We do not
know what motives or modular forms we can expect.
Again, for the cohomology of \({\mathcal {M}}_{g,n}\)  we know that we are dealing with
motives or modular forms `of level one', or in other words,
with everywhere good reduction.
As far as I know, nobody has an idea of the nature of
the zeta function of \({\mathcal {M}}_{g}\)  for general \(g\) .
Results of
Chenevier, Lannes and Renard
on motives of level 1 and small rank
[1]}, [2]} can help
in identifying the results for low values of \(g\)  and \(n\) .
